Robust low-rank matrix completion via sparsity-inducing regularizer

This paper proposes a sparsity-inducing regularizer associated with the Welsch function. We theoretically show that the regularizer is quasiconvex and the corresponding Moreau envelope is convex. Moreover, the closed-form solution to its Moreau envelope, namely, the proximity operator, is derived. Unlike conventional nonconvex regularizers like the ${\ell }_p$-norm with $0<p<1$ that generally needs iterations to obtain the corresponding proximity operator, the developed regularizer has a closed-form proximity operator. We utilize our regularizer to penalize the singular values as well as sparse outliers of the distorted data, and develop an efficient algorithm for robust matrix completion. Convergence of the suggested method is analyzed and we prove that any accumulation point is a stationary point. Finally, experimental results demonstrate that our algorithm is superior to the competing techniques in terms of restoration performance. MATALB codes are available at https://github.com/bestzywang/RMC-NNSR.